We study hypergraphs which represent finite quantum event structures. We contribute to results of graph theory, regarding bounds on the number of edges, given the number of vertices. We develop a missing one for 3-graphs of girth 4. As an application of the graph-theoretical approach to quantum structures, we show that the smallest orthoalgebra with an empty state space has 10 atoms. Optimized constructions of an orthomodular poset and an orthomodular lattice with no group-valued measures are given. We present also a handcrafted construction of an orthoalgebra with no group-valued measure; it is larger, but its properties can be verified without a computer.

中文翻译：

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无状态量子结构和极值图论

我们研究表示有限量子事件结构的超图。我们为图论的结果做出贡献，考虑到顶点数，边数的界限。我们为周长为 4 的 3 图开发了一个缺失的图。作为图论方法对量子结构的应用，我们展示了具有空状态空间的最小正代数有 10 个原子。给出了无群值测度的正交调偏偏集和正交模格的优化构造。我们还展示了一个没有群值测度的正代数的手工构造；它较大，但无需计算机即可验证其属性。